Converters with higher pulse-count include 12-, 18- and 24-pulse configurations, which are generally considered to improve the quality of DC voltage and current at the output terminals as well as of the input AC current. To demonstrate the generalization of the analyses presented above for the conventional three-phase six-pulse converter, an example six-phase twelve-pulse rectifier shown in Figure 10.24 is considered. A similar configuration may be achieved by utilizing two sets of wye/delta windings of a conventional three-phase system (transformer and/or synchronous generator). Without loss of generality, the two six-pulse bridges form a parallel connection. The displacement angle between the two three- phase sets is commonly chosen to be 30 electrical degrees, but in certain applications the displacement angle may be 60 electrical degrees [59]. The configuration of the 12-pulse rectifier may also be varied by either including or excluding the interphase transformer (IPT) and by connecting/disconnecting the
ia1s
vdc1 +
– S1
va1s +
– S2 S3
S4 S5 S6 –+
idc1
v+ab1_s
–
RL Load
–+ –+
LL ec1s
eb1s ea1s Lc
Lc Lc
ia
2 s
vdc2 +
– S7
va2s +
– S8 S9
S10 S11 S12 –+
idc2
v+ab2_s
–
ec2s eb2s ea2s Lc
Lc Lc
idc
vdc +
– –+
–+
Figure 10.24 Typical six-phase twelve-pulse bridge rectifier system.
Table 10.9 Operational modes of the 12-pulse rectifier.
Operational modes Conduction pattern
1 4–2
2 5–4–2–4
3 5–4
4 6–5–4–5
5 6–5
6 6
7 7–6
neutral points of the two sets of three-phase voltage sources. These configurations result in a more complicated switching pattern and a large number of operational modes which are more difficult to establish analytically [59, 60].
10.6.1 Detailed Analysis
Modes of operation for the case of 30◦ displacement angle, disconnected neutrals and without the interphase transformer have been analytically established in [33], where a simplified case of constant DC bus current is assumed. These modes are summarized in Table 10.9, and the regulation characteristic for this case is shown in Figure 10.25. Reference [33] also assumes a case with an ideal interphase
Mode 6
Mode 7
Mode 2
Mode 5
Mode 4
Mode 3
Mode 1
0.22 0.94 1
0 0.1 0.57
0 0.2 0.4 0.6 0.8 1
Id Id0,6 Vd
Vd0,6
Figure 10.25 Steady-state regulation characteristic for the six-phase twelve-pulse bridge converter with the neutral points disconnected.
Table 10.10 Operational modes of the 12-pulse rectifier with ideal IPT.
Operational modes Conduction pattern
1 4–2
2 5–4–2–4
3 5–4
transformer, that is, the magnetizing reactance of the IPT is assumed to be infinite. Thus the load current will be equally shared between the two bridges, which operate independently. Under these assumptions, the operational modes may be derived by analysing one of the bridges with one-half of the load current [33]. Three modes of operation are then recognized, which are summarized in Table 10.10. It should be noted that in the case of a non-ideal interphase transformer, the regulation characteristic will lie between these two extreme cases [33].
If the neutral points of the two sets of three-phase voltage sources in Figure 10.24 are connected, a new set of line-to-line voltages is established between the phases. This will allow the phase current waveforms to become asymmetric, resulting in more complicated operational modes. The regulation characteristic for this case is shown in Figure 10.26, with the modes of operation summarized in Table 10.11.
10.6.2 Dynamic Average Modelling
The dynamic AVM for the above 12-pulse rectifier system has been developed using the generalized parametric approach [27]. Simulation studies of the twelve-pulse rectifier have been conducted using the
Figure 10.26 Steady-state regulation characteristic for the six-phase twelve-pulse bridge converter with connected neutral points.
Table 10.11 Operational modes of the 12-pulse rectifier with connected neutral points.
Operational modes Conduction pattern
1 3–4–3–2
2 5–4–3–2–3–4
3 5–4–3–3–2–3–4
4 5–4–3–4–3–3–2–3–4
5 5–4–3–4–3–3–4
6 5–4–3–4–3–4–3–4
7 5–4–3–4–4–3–4
8 5–4–3–4–4
9 5–4–5–4–3–4
10 5–4–5–4–4
11 5–4
12 6–5–4–5–4–5
13 6–5–4–5
14 6–5–5
15 6–5–6–5
16 6–6–5
17 6
18 7–6
detailed model and the parametric AVM only, because the analytically derived models do not capture the intermode transitions. A transient study has been carried out in which the load resistance is stepped from 1Ωto 0.1Ωatt=0.5 s. The corresponding responses, for the case of connected neutral points, are superimposed in Figure 10.27. In this case, the operational mode is changing from Mode 13 (i.e.
6–5–4–5 conduction pattern) to Mode 17 (6-valve conduction pattern). As the plots of this Figure 10.27 show, the AVM predicts the transient response very accurately.
To evaluate the effectiveness of the AVMs relative to the switching models, we can compare the time- step size and the total number of time-steps that were required by each of the models to complete the entire transient response. For the purpose of comparison, a transient study duration of 1 s was assumed.
For example, in case of six-pulse rectifier the time-steps taken by each of the models implemented in Simulink are summarized in Table 10.12. All Simulink models were executed using a variable time-step solver that can automatically adjust the step size during the transient. The table shows that the switching model required the largest number of time-steps (22 659), which were needed to accurately handle all the switching events (discontinuities). The AVMs could utilize a much larger time-step, because these models are continuous, taking far fewer steps (281, 271 and 309, respectively).
The studies with 12-pulse rectifier were carried out using both PSCAD and Simulink. The summary of the time-steps is also given in Table 10.12. For the considered time interval/study of 1 s, the detailed model again took the largest number of steps (20 001 and 15 366). There is some difference between the PSCAD and Simulink detailed models, which can be attributed to the fact that PSCAD uses fixed time- step to solve the entire transient, whereas Simulink can vary the time- step to accommodate the switching and other transients. The detailed PSCAD model was run with a typical time-step of 50μs required to properly handle the switching of diodes. However, the PSCAD-AVM and Simulink-AVM could use appreciably larger time-steps, which altogether demonstrates the benefits of the AVM approach, where each model took significantly fewer time-steps (5586 and 194). The PSCAD-AVM could not run at very large time-steps because the time-step was still limited by the relatively fast transient observed during the rapid change in the load.
14
2 20
vdc (V)
0.495 0.510 0.525
Detailed
Parametric AVM
60
0 120
idc (A)
0.495 0.510 0.525
Detailed Parametric AVM
25
10 40
iqs (A)
Detailed
Parametric AVM
30 60
ids (A) Detailed
Parametric AVM
0
0.495 0.510 0.525
0.495 0.510 0.525
Time (s) 8
Figure 10.27 Twelve-pulse bridge converter transient response from Mode 13 to Mode 17 as predicted by detailed and averaged models.
Table 10.12 Comparison of model performance.
Rectifier Model Time-steps
Six-pulse rectifier Detailed model 22 659
AVM-1 281
AVM-2 271
Parametric AVM 309
Twelve-pulse rectifier Detailed – PSCAD 20 001
Detailed – Simulink 15 366
AVM – PSCAD 5586
AVM – Simulink 194
Dynamic average models are extremely advantageous for simulations of system transients where the switching harmonics injected into the AC grid or the DC link are neglected. Including the effect of switching harmonics would require a special consideration and may be pursued in combination with other approaches, such as multiple reference frames and harmonic-domain modelling. Dynamic AVMs neglect (remove) the effect of fast switching but preserve the slower dynamics of the system. As a result, they can be effectively used to improve the simulation efficiency for the system-level transient studies with a large number of such loads and subsystems.